OCCUPATION TIME DISTRIBUTIONS FOR LEVY BRIDGES AND EXCURSIONS

Let X be a one-dimensional Levy process. It is shown that under the bridge law for X starting from 0 and ending at 0 at time t, the amount of time X spends positive has a uniform distribution on [O, t]. When O is a regular point, this uniform distribution result leads to an explicit expression for the Laplace transform of the joint distribution of the pair (R, A(R)), where R is the length of an excursion of X from 0, and A(R) is the total time X spends positive during the excursion. More concrete expressions are obtained for stable processes by specialization. In particular, a formula determining the distribution of A(R)/R is given in the stable case.